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- 1. 9.10.11.12.13 June 2014 Singapore Mathematics Institute with Dr. Yeap Ban Har coursebook
- 2. 2 | P a g e Contact Information yeapbanhar@gmail.com www.banhar.blogspot.com about yeap ban har Dr Yeap Ban Har spent ten years at Singapore's National Institute of Education training pre-service and in-service teachers and graduate students. Ban Har has authored dozens of textbooks, math readers and assorted titles for teachers. He has been a keynote speaker at international conferences, and is currently the Principal of a professional development institute for teachers based in Singapore. He is also Director of Curriculum and Professional Development at Pathlight School, a primary and secondary school in Singapore for students with autism. In the last month, he was a keynote speaker at World Bank’s READ Conference in St Petersburg, Russia where policy makers from eight countries met to discuss classroom assessment. He was also a visiting professor at Khon Kaen University, Thailand. He was also in Brunei to work with the Ministry of Education Brunei on a long-term project to provide comprehensive professional development for all teachers in the country.
- 3. 3 | P a g e introduction The Singapore approach to teaching and learning mathematics was the result of trying to find a way to help Singapore students who were mostly not performing well in the 1970’s. The CPA Approach as well as the Spiral Approach are fundamental to teaching mathematics in Singapore schools. The national standards, called syllabus in Singapore, is designed based on Bruner’s idea of spiral curriculum. Textbooks are written based on and teachers are trained to use the CPA Approach, based on Bruner’s ideas of representations. “A curriculum as it develops should revisit this basic ideas repeatedly, building upon them until the student has grasped the full formal apparatus that goes with them”. | Bruner 1960 “I was struck by the fact that successful efforts to teach highly structured bodies of knowledge like mathematics, physical sciences, and even the field of history often took the form of metaphoric spiral in which at some simple level a set of ideas or operations were introduced in a rather intuitive way and, once mastered in that spirit, were then revisited and reconstrued in a more formal or operational way, then being connected with other knowledge, the mastery at this stage then being carried one step higher to a new level of formal or operational rigour and to a broader level of abstraction and comprehensiveness. The end stage of this process was eventual mastery of the connexity and structure of a large body of knowledge.” | Bruner 1975 Bruner's constructivist theory suggests it is effective when faced with new material to follow a progression from enactive to iconic to symbolic representation; this holds true even for adult learners. | Bruner 1966
- 4. 4 | P a g e Open Lesson | What do we want the students to learn? Lesson Segment Observation / Question How can we tell if students are learning? What help students who struggle? What are for students who already know what we want them to learn? Summary
- 5. 5 | P a g e Day 1 | Early Numeracy |Session 1 Rational Counting Number Bonds Lesson Sequence Use of Literature Lesson Sequence Anchor Task Guided Practice (Independent Practice) Case Study 1 | Show 5 beans on a ten frame. Do it in another way.
- 6. 6 | P a g e Case Study 2 | Show the teacher five pieces of square tiles. Make a shape using five square tiles. There are some rules that we have to follow.
- 7. 7 | P a g e Whole Number Addition and Subtraction |Session 2 Materials Strategies Semantics Variation Semantics Part-Whole Change Comparison Case Study 2 | Together, Jon and Kim have 32 coins. Jon has 19 coins. Find the number of coins that Kim has.
- 8. 8 | P a g e Case Study 3 | Lance has 10 coins more than Ming. Together, they have 34 coins. How many coins does Lance have?
- 9. 9 | P a g e Case Study 4 | At first, Lance had 10 coins more than Ming. Then Ming gave Lance 6 coins. Who had more coins in the end? How many more?
- 10. 10 | P a g e Use of Activities for Math Learning |Session 4 Types of Lessons To develop basic ideas, concepts and skills To consolidate basic ideas, concepts and skills To extend basic ideas, concepts and skills Case Study 5 | Use the digits 0 to 9 not more than once to make an addition equation.
- 11. 11 | P a g e Holistic Assessment for Young Learners |Session 5 Assessment Benchmarks Approaching Expectations Meeting Expectations Exceeding Expectations Students should be able to perform rational counting. Approaching Expectations The student is unable to count a plate of not more than ten cookies. Can the student perform one to one correspondence? Can the student classify? Can the student rote count? Has the student grasp the principle of cardinality? Meeting Expectations The student is able to count a plate of not more than ten cookies. Also able to read the correct numeral Also able to read the correct number word Also able to write the correct numeral Also able to write the correct number word Exceeding Expectations The student is able to count a plate of not more than ten cookies. The student is also able to read and write the correct numeral and number word.
- 12. 12 | P a g e Day 2 | Differentiated Instruction |Session 1 and Session 2 Remediation Enrichment Four Critical Questions Four Critical Questions (DuFour) What do I want the students to learn? How do I know they have learnt it? What if they cannot learn it? What if they already learnt it? Differentiated Instruction (Tomlinson) Content Process Product Affect
- 13. 13 | P a g e Case Study 1 | Basic Idea Lesson Draw any triangle. How are the three angles in a triangle related? Answer the four critical questions. DI for Struggling Learners DI for Advanced Learners
- 14. 14 | P a g e Case Study 2 | Basic Idea Lesson Anchor Task | Mom baked two cakes. After giving half of a cake to our neigbors, we ate 5 4 of a cake. Answer the four critical questions. DI for Struggling Learners DI for Advanced Learners
- 15. 15 | P a g e Case Study 3 | Practice Lesson Draw triangles and find the area of each. Answer the four critical questions. DI for Struggling Learners DI for Advanced Learners
- 16. 16 | P a g e Use of Games in Math Learning |Session 4 Types of Lessons To develop basic ideas, concepts and skills To consolidate basic ideas, concepts and skills To extend basic ideas, concepts and skills Case Study 4 | Write expressions that include fractions and one of the four basic operations, one on each side of the square such that the value of adjacent expressions are equal in value. Cut out the pieces, mix them up and ask another group to arrange the pieces back again such that values of adjacent expressions are equal.
- 17. 17 | P a g e Journal Writing |Session 5 Case Study 5 | Problem-Solving Lesson Let’s have a go at writing a math journal using this diagram as a stimulus.
- 18. 18 | P a g e Day 3 | Whole Number Multiplication and Division |Session 1 Strategies Semantics Multiplication Group Array Area Rate Combination Division Sharing Grouping Case Study 1 | Compare these three lessons on division of whole numbers Anchor Task A | Try putting 14 children into 3 equal groups.
- 19. 19 | P a g e Anchor Task B | Try putting 41 children into groups of threes. Anchor Task C | Try putting 41 liters of water into 3 containers. Is it possible to make sure each container contains the same amount of water?
- 20. 20 | P a g e Case Study 2 | X = Given three digits, make two numbers, a 1-digit number and a 2-digit number, so that the product has the largest possible value.
- 21. 21 | P a g e Factors and Multiples |Session 2 Jerome Bruner Zoltan Dienes Richard Skemp Case Study 3 | Use 12 square tiles to make a rectangle or square.
- 22. 22 | P a g e Model Drawing |Session 4 Case Study 4 | There are 40 children is Primary 3 Honesty. 19 of them are boys. How many girls are there in Primary 3 Honesty? Case Study 5 | There are three times as many boys as there are girls in the soccer club. There are 96 children in the soccer club. Is this possible?
- 23. 23 | P a g e Case Study 6 | There is a group of people in a room. A third of them are children. A third of the children are boys. There are 9 or 10 children in the room. Which situation is possible? For that situation, suggest questions that can be answered using the given information.
- 24. 24 | P a g e Holistic Assessment |Session 5 Newman’s Procedure o Read o Comprehend o Know Strategies o Transform o Compute o Interpret Approaching Expectations Unable to solve word problems that is required at the current grade level. However, the student is able to handle single-step word problems. Meeting Expectations Able to handle typical word problem required at the current grade level. Exceeding Expectations Able to handle unusual word problems and / or complex word problems. Case Study 7 | At first, the ratio of the number of students in Basketball to the number of students in Soccer was 3 : 1. When 18 students moved from Basketball to Soccer, the there were equal number of students in both sports. Find the number of students in these two sports. What if the ratio is 4 : 1?
- 25. 25 | P a g e Case Study 8 | In a group of 96 students, a third of the boys and a fifth of the girls do not have pets at home while 70 students have pets at home. How many boys have pets at home?
- 26. 26 | P a g e Fractions, Fractions, Fractions! an in-depth study of the teaching of fractions Day 4 | Two Fundamentals |Session 1 Idea of ‘Nouns’ Idea of Equal Parts Case Study 1 |
- 27. 27 | P a g e Show 2 equal parts. What do you mean by equal parts? Show 4 equal parts. In lesson study, we might discuss why use squares. Why not circles? Why not rectangles?
- 28. 28 | P a g e Case Study 2 | 5 2 5 1 5 1 5 3 Equivalent Fractions |Session 2 Case Study 3 | 8 ? 4 1 ? 9 4 3
- 29. 29 | P a g e A cake is cut into 6 equal slices. Aaron and Ben share four slices. Case Study 4 | What fraction of the rectangle is shaded?
- 30. 30 | P a g e Basic Operations |Session 4 Case Study 5 | Mary has a blue ribbon that is 3 2 1 m long. She has a red ribbon that is 4 3 1 m long. Source | Primary Mathematics (Standards Edition) 6A
- 31. 31 | P a g e Case Study 6 | There are 12 cupcakes left over. Alex takes 4 3 of them home. How many cupcakes does Alex take? There are 2 1 3 pies left over. Ali takes 4 3 of them home. How many pies does Ali take? Source | Primary Mathematics (Standards Edition) 6A
- 32. 32 | P a g e Case Study 7 | The longest side of a triangle is 4 3 2 times as long as the shortest side. The shortest side is 3 2 in. Find the length of the longest side. Source | Primary Mathematics (Standards Edition) 6A
- 33. 33 | P a g e Practice and Problem Solving |Session 5 Instructional Models o Teaching through Problem Solving o Teaching for Problem Solving o Teaching of Problem Solving Case Study 8 | A total of 325 boys and girls attended a performance in the school hall. 5 4 of the boys and 4 3 of the girls left the hall after the performance ended. There were 29 more boys than girls who remained in the hall. How many girls attended the performace? Source | Catholic High School (Primary) Primary 6 Examination
- 34. 34 | P a g e Case Study 9 |
- 35. 35 | P a g e Day 5 | Ratio and Proportion |Session 1 Problem-Solving Approach Three-Part Lesson Format A total of 325 boys and girls attended a performance in the school hall. 5 4 of the boys and 4 3 of the girls left the hall after the performance ended. There were 29 more boys than girls who remained in the hall. How many girls attended the performace? Source | Catholic High School (Primary) Primary 6 Examination
- 36. 36 | P a g e Case Study 1 | Find the area of a polygon with one dot inside it. How does the area vary with the number of dots on the perimeter of the polygon?
- 37. 37 | P a g e Find the area of a polygon with four dots on the perimeter. How does the area vary with the number of dots inside the polygon?
- 38. 38 | P a g e Advanced Bar Model Method |Session 2 Case Study 2 | Four friends, Ravi, Johan, Meng and Emma, shared the cost of a present. Ravi paid 50% of the total amount paid by the other three friends. Meng paid 60% of the total amount paid by Johan and Emma. Johan paid ½ of what Emma paid. Ravi paid $24 more than Emma. How much did the present cost? Source | Primary Six Examination in a Singapore School
- 39. 39 | P a g e Case Study 3 | At a swimming meet, School A had 18 more swimmers than School B and 6 fewer swimmers than School C. The ratio of the number of boys to the number of girls from the three schools was 1 : 3. The ratio of the number of boys to the number of girls in School A, School B and School C were 1 : 3, 1 : 5 and 2 : 5, respectively. Find the total number of swimmers from the three schools. Source | Primary Six Examination in a Singapore School
- 40. 40 | P a g e Teaching Algebra |Session 4 Ideas Development o Variable o Expression Simplify Expand Factor o Equation Linear Quadratic Others Case Study 4 | Solve 7 – x = 4. Source | Primary Mathematics (Standards Edition) 6A Case Study 5 | There are three times as many boys as there are girls in the soccer club. There are 96 children in the soccer club. Number of boys Number of girls
- 41. 41 | P a g e Case Study 6 | (a) Find the value of 3s – 1 when s = 4. (b) Solve 3s – 1 = 11. Source | Primary Mathematics (Standards Edition) 6A Case Study 7 | Is it possible to factor 252 2 xx into linear factors?
- 42. 42 | P a g e Is it possible for 252 2 xx = 0? Case Study 8 | Use algebra tiles to show 522 xx and 142 xx . In each case try to rearrange the tiles to form a square.
- 43. 43 | P a g e Holistic Assessment |Session 5 Skemp’s Types of Understanding o Instrumental o Relational o Conventional Approaching Expectations Student is unable to solve typical systems of linear equations. The source of difficulty is likely to be knowing the meaning of ‘solve’ (conventional) knowing how to read algebraic expressions (conventional) knowing how to do arithmetic manipulation (instrumental) … Meeting Expectations Student is able to solve typical systems of linear equations. Exceeding Expectations Student is able to solve typical systems of linear equations. There is also evidence that the student is able to extend his/her understanding to less common situations. Case Study 9 | Solve 171 2 1 3 1 3 1 2 1 yxyx .

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